Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 482840, 8 pages
doi:10.1155/2011/482840
Research Article
Some Identities on the Twisted
h, q-Genocchi Numbers and Polynomials
Associated with q-Bernstein Polynomials
Seog-Hoon Rim1 and Sun-Jung Lee2
1
Department of Mathematics Education, Kyungpook National University,
Daegu 702-701, Republic of Korea
2
Department of Mathematics, Kyungpook National University, Daegu 702-701, Republic of Korea
Correspondence should be addressed to Seog-Hoon Rim, shrim@knu.ac.kr
Received 9 October 2011; Accepted 15 November 2011
Academic Editor: Taekyun Kim
Copyright q 2011 S.-H. Rim and S.-J. Lee. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We give some interesting identities on the twisted h, q-Genocchi numbers and polynomials
associated with q-Bernstein polynomials.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, we always make use of the
following notations: Z denotes the ring of rational integers, Zp denotes the ring of padic rational integer, Qp denotes the ring of p-adic rational numbers, and Cp denotes the
completion of algebraic closure of Qp , respectively. Let N be the set of natural numbers and
n
Z N {0}. Let Cpn {ζ | ζp 1} be the cyclic group of order pn and let
Tp n≥1
Cpn lim Cpn Cp∞ .
n→∞
1.1
The p-adic absolute value is defined by |x| 1/pr , where x pr s/t r ∈ Q and s, t ∈ Z
with s, t p, s p, t 1. In this paper we assume that q ∈ Cp with |q − 1|p < 1 as an
indeterminate.
2
International Journal of Mathematics and Mathematical Sciences
The q-number is defined by
xq 1 − qx
1−q
1.2
see 1–15. Note that limq → 1 xq x. Let UDZp be the space of uniformly differentiable
function on Zp . For f ∈ UDZp , Kim defined the fermionic p-adic q-integral on Zp as follows:
I−q f −1
p
x
1
fxdμ−q x lim N fx −q
N
→
∞
p −q x0
Zp
N
1.3
see 2–6, 8–15. From 1.3, we note that
n−1
qn I−q fn −1n I−q f 2q −1n−1− q f
1.4
0
see 4–6, 8–12, where fn x fx n for n ∈ N. For k, n ∈ Z and x ∈ 0, 1, Kim defined
the q-Bernstein polynomials of the degree n as follows:
Bk,n x, q n
k
xkq 1 − xn−k
q−1 ,
1.5
see 13–15. For h ∈ Z and ζ ∈ Tp , let us consider the twisted h, q-Genocchi polynomials
as follows:
t
∞
tn
h
exyq t ζy qh−1y dμ−q y Gn,q,ζ x .
n!
Zp
n0
1.6
h
Then, Gn,q,ζ x is called nth twisted h, q-Genocchi polynomials.
h
h
In the special case, x 0 and Gn,q,ζ 0 Gn,q,ζ are called the nth twisted h, q-Genocchi
numbers.
In this paper, we give the fermionic p-adic integral representation of q-Bernstein
polynomial, which are defined by Kim 13, associated with twisted h, q-Genocchi numbers
and polynomials. And we construct some interesting properties of q-Bernstein polynomials
associated with twisted h, q-Genocchi numbers and polynomials.
International Journal of Mathematics and Mathematical Sciences
3
2. On the Twisted h, q-Genocchi Numbers and Polynomials
From 1.6, we note that
h
Gn1,q,ζ x
n1
Zp
n
x y q ζy qh−1y dμ−q y
n
xq qx y q ζy qh−1y dμ−q y
Zp
n
n
0
x
xn−
q q
n
n
h
x
xn−
q q
0
Zp
2.1
y h−1y
y qζ q
dμ−q y
G1,q,ζ
1
.
We also have
h
Gn,q,ζ x
q
−x
n
n
0
h
x
xn−
q q G,q,ζ .
2.2
Therefore, we obtain the following theorem.
Theorem 2.1. For n ∈ Z and ζ ∈ Tp , one has
h
h n
Gn,q,ζ x q−x xq qx Gq,ζ
2.3
h n
with usual convention about replacing Gq,ζ by Ghn,q,ζ .
By 1.6 and 2.1 one gets
h
Gn1,q−1 ,ζ−1 1 − x
n1
Zp
n
1 − x y q−1 ζ−y q−h−1y dμ−q−1 y
2q
1−q
−1 n
n
n
0
−1n qnh−1 ζ
−1n qh−1 ζ
2q
1−q
n
n
n
0
h
−1n ζqnh−1
Gn1,q,ζ x
n1
Therefore, we obtain the following theorem.
qx
1 qh ζ
.
−1
x
q
1 qh ζ
2.4
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International Journal of Mathematics and Mathematical Sciences
Theorem 2.2. For n ∈ Z and ζ ∈ Tp , one has
h
h
Gn,q−1 ,ζ−1 1 − x −1n−1 ζqnh−2 Gn,q,ζ .
2.5
From 1.5, one gets the following recurrence formula:
h
h
qh ζGn,q,ζ 1 Gn,q,ζ ⎧
⎨2q
if n 1,
⎩0
if n > 1.
2.6
Therefore, we obtain the following theorem.
Theorem 2.3. For n ∈ Z and ζ ∈ Tp , one has
G0,q,ζ 0,
qh−1 ζ
h
qGq,ζ
⎧
⎨2q
n
h
1 Gn,q,ζ ⎩0
if n 1,
if n > 1
2.7
h n
with usual convention about replacing Gq,ζ by Ghn,q,ζ .
From Theorem 2.3, we note that
2h 2
q ζ
h
Gn,q,ζ 2
− q ζn2q −q
h
h−1
ζ
n
n
0
h
q G,q,ζ
n
h
−qh−1 ζ qGq,ζ 1
h
Gn,q,ζ
2.8
if n > 1.
Therefore, we obtain the following theorem.
Theorem 2.4. For n ∈ Z and ζ ∈ Tp , one has
h
h
q2h ζ2 Gn,q,ζ 2 Gn,q,ζ nqh ζ2q .
2.9
Remark 2.5. We note that Theorem 2.4 also can be proved by using fermionic integral equation
1.4 in case of n 2.
International Journal of Mathematics and Mathematical Sciences
5
By 2.4 and Theorem 2.2, we get
h
Gn1,q−1 ,ζ−1 2
n1
h
n nh−1
−1 q
ζ
Gn1,q,ζ −1
n1
−1n qnh−1 ζ
q
h−1
ζ
Zp
Zp
x − 1nq ζx qh−1x dμ−q x
2.10
1 − xnq−1 ζx qh−1x dμ−q x.
Therefore, we obtain the following theorem.
Theorem 2.6. For n ∈ Z and ζ ∈ Tp , one has
n 1qh−1 ζ
h
Zp
1 − xnq−1 ζx qh−1x dμ−q x Gn1,q−1 ,ζ−1 2.
2.11
Let n ∈ N. By Theorems 2.4 and 2.6, we get
n 1q
h−1
ζ
h
Zp
1 − xnq−1 ζx qh−1x dμ−q x q2h ζ2 Gn1,q−1 ,ζ−1 n 1qh−1 ζ2q .
2.12
Therefore, we obtain the following corollary.
Corollary 2.7. For n ∈ Z and ζ ∈ Tp , one has
Zp
h
1 −
xnq−1 ζx qh−1x
dμ−q x q
h1
ζ
Gn1,q−1 ,ζ−1
n1
2q .
2.13
By 1.5, we get the symmetry of q-Bernstein polynomials as follows:
Bk,n x, q Bn−k,n 1 − x, q−1
(see [11]).
2.14
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International Journal of Mathematics and Mathematical Sciences
Thus, by Corollary 2.7 and 2.14, we get
Zp
Bk,n x, q qh−1x ζx dμ−q x Zp
Bn−k,n 1 − x, q−1 qh−1x ζx dμ−q x
k
k
n k−
h−1x x
ζ dμ−1 x
1 − xn−
−1
q−1 q
k 0 Zp
⎛
⎞
h
k
Gn−1,q−1 ,ζ−1
k
n 2q ⎠
−1k− ⎝qh1 ζ
n−1
k 0 ⎧
h
⎪
Gn1,q−1 ,ζ−1
⎪
⎪
h1
⎪
2q
q ζ
⎪
⎪
⎨
n1
⎛ ⎞ ⎛ ⎞
h
⎪
k
Gn−1,q−1 ,ζ−1
k
⎪
⎪ h1 ⎝n⎠
k−
⎪
⎝ ⎠−1
⎪
q ζ
⎪
⎩
n−1
k 0 2.15
if k 0,
if k > 0.
From 2.15, we have the following theorem.
Theorem 2.8. For n ∈ Z and ζ ∈ Tp , one has
⎧
h
⎪
Gn1,q−1 ,ζ−1
⎪
⎪
h1
⎪
2q
q ζ
⎪
⎪
⎨
n1
h−1x x
⎛ ⎞ ⎛ ⎞
Bk,n x, q q
ζ dμ−q x h
⎪
k
G
k
n ⎪
Zp
−1 −1
⎪
k− n−1,q ,ζ
h1
⎪
⎝
⎠
⎝
⎠
⎪
q ζ
−1
⎪
⎩
n−1
k 0 if k 0,
2.16
if k > 0.
For n, k ∈ Z with n > k, fermionic p-adic invariant integral for multiplication of two qBernstein polynomials on Zp can be given by the following:
Zp
Bk,n x, q q
h−1x x
ζ dμ−q x n
Zp
h−1x x
ζ dμ−q x
xkq 1 − xn−k
q−1 q
k
n
n−k
qh−1x ζx dμ−1 x
xkq 1 − xq
k
n−k n − k
n Zp
k
0
−1
Zp
h−1x x
ζ dμ−1 x.
xk
q q
From Theorem 2.8 and 2.17, we have the following corollary.
2.17
International Journal of Mathematics and Mathematical Sciences
7
Corollary 2.9. For n ∈ Z and ζ ∈ Tp , one has
⎧
h
⎪
Gn1,q−1 ,ζ−1
⎪
⎪
h1
⎪
2q
q ζ
⎪
h
⎪
n−k n − k
⎨
n1
G
k1,q,ζ
⎛ ⎞
−1
h
k1 ⎪
k
Gn−1,q−1 ,ζ−1
k
⎪
0
⎪
k−
h1
⎪q ζ ⎝ ⎠−1
⎪
⎪
⎩
n−1
0
if k 0,
2.18
if k > 0.
Let n1 , n2 , k ∈ Z with n1 n2 > 2k. Then we get
Bk,n1 x, q Bk,n2 x, q qh−1x ζx dμ−q x
Zp
2k
2k
n2
n1
k
k
1 n2 − h−1x x
q
ζ dμ−q x
−12k− 1 − xnq−1
Zp 0
⎛
2k
2k
n2 n1
k
k
−1
0
2k− ⎝
h
−1 −1
1 n2 −1,q ,ζ
Gn
n1 n2 − 1
2.19
⎞
q
h1
ζ 2q ⎠.
From 2.19, we have the following theorem.
Theorem 2.10. For n ∈ Z and ζ ∈ Tp , one has
Zp
Bk,n1 x, q Bk,n2 x, q qh−1x ζx dμ−q x
⎧
h
⎪
Gn n 1,q−1 ,ζ−1
⎪
1
2
⎪
h1
⎪
2q
q ζ
⎪
⎪
⎨
n1 n2 1
⎛ ⎞⎛ ⎞ ⎛ ⎞
h
⎪
2k
G
n1
n2 2k
⎪
−1 −1
⎪
2k− n1 n2 −1,q ,ζ
⎪
⎝
⎠
⎝
⎠
⎝
⎠
⎪
−1
⎪
⎩
n1 n2 − 1
k
k 0 if k 0,
2.20
if k > 0.
Let n1 , n2 , k ∈ Z with n1 n2 > 2k, fermionic p-adic invariant integral for multiplication of
two q-Bernstein polynomials on Zp can be given by the following:
Zp
Bk,n1 x, q Bk,n2 x, q qh−1x ζx dμ−q x
n2
n1
k
k
n1 n
2 −2k
Zp
k
0
−1
n1 n2 − 2k
0
n n −2k
2
n1
n2 1 k
−1
n1 n2 − 2k
qh−1x ζx dμ−q x
x2k
q
h
G2k1,q,ζ
2k 1
.
From Theorem 2.10 and 2.21, we have the following corollary.
2.21
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International Journal of Mathematics and Mathematical Sciences
Corollary 2.11. For n1 , n2 , k ∈ Z and n1 n2 > 2k, one has
n1 n
2 −2k
0
n1 n2 − 2k
h
−1
G2k1,q,ζ
2k 1
⎧
h
⎪
Gn n 1,q−1 ,ζ−1
⎪
1
2
⎪
h1
⎪
2q
q ζ
⎪
⎪
⎨
n1 n2 1
⎞
⎛
h
⎪
n1 n
G
2 −2k
n1 n2 − 2k
⎪
−1 −1
⎪
2k− n1 n2 −1,q ,ζ
⎪
⎠
⎝
⎪
−1
⎪
⎩
n1 n2 − 1
0
if k 0,
2.22
if k > 0.
Acknowledgment
The authors would like to thank the anonymous referee for his/her excellent detail comments
and suggestion.
References
1 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
2 T. Kim, “A note on p-adic q-integral on Zp Associated with q-Euler numbers,,” Advanced Studies in
Contemporary Mathematics , vol. 15, no. 2, pp. 133–138, 2007.
3 T. Kim, “A note on q-Volkenborn integration,” Proceedings of the Jangjeon Mathematical Society, vol. 8,
no. 1, pp. 13–17, 2005.
4 T. Kim, “On the multiple q-Genocchi and Euler numbers,” Russian Journal of Mathematical Physics, vol.
15, no. 4, pp. 481–486, 2008.
5 T. Kim, “On the q-extension of Euler and Genocchi numbers,” Journal of Mathematical Analysis and
Applications, vol. 326, no. 2, pp. 1458–1465, 2007.
6 T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the
fermionic p-adic integral on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491,
2009.
7 T. Kim, “Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral
on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009.
8 T. Kim, “On the multiple q-Genocchi and Euler numbers,” Russian Journal of Mathematical Physics, vol.
15, no. 4, pp. 481–486, 2008.
9 H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with
multiple q-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol. 12, no. 2,
pp. 241–268, 2005.
10 I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order w-q-Genocchi numbers,”
Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.
11 R. Dere and Y. Simsek, “Genocchi polynomials associated with the Umbral algebra,” Applied
Mathematics and Computation, vol. 218, no. 3, pp. 756–761, 2011.
12 H. Ozden, I. N. Cangul, and Y. Simsek, “A new approach to q-Genocchi numbers and their
interpolation functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e793–e799, 2009.
13 T. Kim, “A note on q-Bernstein polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 1, pp.
73–82, 2011.
14 L. C. Jang, W. J. Kim, and Y. Simsek, “A study on the p-adic integral representation on Zp associated
with Bernstein and Bernoulli polynomials,” Advances in Difference Equations, vol. 2010, Article ID
163217, 6 pages, 2010.
15 D. V. Dolgy, D. J. Kang, T. Kim, and B. Lee, “Some new identities on the twisted h, q-Euler numbers
and q-Bernstein polynomials,” Journal of Computational Analysis and Applications. In press.
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